Transform one object to another with animation
This is what I mean in my third comment, ie: if you can use "text" objects (not letters meshes), you could also start from the "straight" text and use the "text on curve" setting, using a curve to achieve the same effect, then you could animate the curve from arc to linear to get the desired effect... basically it's always easier to produce the curve and then reverse the effect in animation... 

here the text object is "deformed" by a "flattened" curve.



You can simply animate the curve deformation in reverse...

If you have text meshes instead, you could use shape keys like this:



And even here you could just animate the shape key values in reverse...

In the end, about other solution I suggested in comments above:


deforming the whole object is not feasible: letters will be distorted
separate single letters and try to animate each from rotated to straight is feasible but extremely difficult, complicated and, imho, not worth. 


I would use one of the methods described above, instead.